I'm a little stuck on a problem about characteristic function. Thanks in advance for any comment, help or suggestions.
Prove that one can find independent random variables $X_1$, $X_2$, $X_3$ such that $\phi_{X_1}(t)\ \phi_{X_3}(t) = \phi_{X_2}(t)\ \phi_{X_3}(t)$ but $\phi_{X_1}(t) \neq \phi_{X_2}(t)$, where $\phi_{X_i}$ for $i = 1,\ 2,\ 3$ represent the corresponding characteristic functions asociated to $X_i$.
Prove there exists a sequence of random variables $\{X_n\}$ and a randon variable $X$ such that $\phi_{X_n}(t) \ \longrightarrow \ \phi_X (t)$ for $|t| \leqslant 1$, but it cannot extend the convergence for $t\in\mathbb{R}$.
For the first one, I have a hint about use the following functions:
- $\phi(t) = \frac{1}{1+t^2} \ \ $ (characteristic function of a r. v. with Laplace distribution)
- It can be shown that a function defined by: $$f(t) = f(-t), \hspace{1cm} f(t+2) = f(t), \hspace{0.5cm} and \hspace{0.5cm} f(t)=1-t, \hspace{0.1cm} for \hspace{0.2cm} 0 \leq t \leq 1, $$ is a characteristic function. So the hint taken from Gnedenko's book (The Theory of probability - Chapter 7 - exercise 6) says that by making use of the 2 characteristic functions presented above, the exercise can be solved, but the truth is that I don't see how to use them well, especially considering that there are 3 characteristic functions requested in the statement 1.
Again, thanks for any idea and suggestion.